Modeling and analysis of networked finite state machine subject to random communication losses
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Volume 1 January 2021 ISSN: 2767-8946
MMC, 3(1): 50–60
DOI:10.3934/mmc.2023005
Mathematical
Mathematical Modelling Modelling and Control
Received: 11 September 2022
Editors in Chief:
and Control Xiaodi Li & Sabri Arik
Revised: 07 January 2023
Accepted: 26 January 2023
http://www.aimspress.com/journal/mmc Published: 15 February 2023
Research article
Modeling and analysis of networked finite state machine subject to random
communication losses
Weiwei Han1 , Zhipeng Zhang2,∗ and Chengyi Xia2,∗
1
Tianjin Key Laboratory of Intelligence Computing and Novel Software Technology, Tianjin University of Technology,
Tianjin, 300384, China
2
School of Artificial Intelligence, Tiangong University, Tianjin, 300387, China
* Correspondence: Email: zpzhang19@126.com, xialooking@163.com.
Abstract: In networked control systems, channel packet loss is inevitable due to the restricted bandwidth, especially in control (from
supervisory controller to some remote actuators), which will lead to the occurrence of failure control. In this paper, the controllability
of networked finite state machine (NFSM) is investigated within the framework of matrix semi-tensor product (STP), where random
channel packet losses are considered. Firstly, to capture the transition dynamics under random packet losses in the control channel,
we introduce a stochastic variable to estimate the state evolution, and the variable is assumed to obey the Bernoulli binary distribution.
Meanwhile, the NFSM with random channel packet losses can be expressed as a probabilistic logic representation. Subsequently, by
means of the delicate operation of matrix STP, some concise validation conditions for the controllability with a probability of one (w.p.
1), are derived for NFSM based on the probabilistic logic representation. Finally, a typical computing instance is used to demonstrate
the validity of the proposed method. The conclusions are conducive to study the security issues of the system involving opacity, fault
detection, controller design and so on.
Keywords: semi-tensor product; networked system; finite state machine; controllability; channel packet losses
1. Introduction makes it is easier for information to be transferred by
means of communication networks; and on the other hand,
the introduction of communication networks renders the
Finite state machine (FSM) has found successful
system with many excellent performances, such as cost
applications in many complex artificial systems, including
reduction, convenient maintenance friendly, easy tests and
network intrusion detection and cyber-physical ones. In
so on. Hence, the control problems of NFSM have attracted
a classical finite state machine, it is assumed that the
increasing attention and has become a very active field.
controller is close to the actuator, the event command can be
transmitted to the controlled plant instantaneously through Generally, due to the relatively far transmission distance,
the control channel. After decades of development, many it is inevitable to introduce delays or packet losses between
problems for such these systems have been well studied, supervisory controller and actuator. Specifically, an event
such as monitor theory [1], state estimate and detectability occurring spontaneously or a control command generated
[2], controllability [3, 4] and so on. It is important to by the monitor could be transferred at random in the
note that [5–7] converting the cyber-physical systems in the model of NFSM. If the monitor is designed without
logical networks and obtained a series of meaningful and considering the delays or packet losses in the control
innovative results about the cyber-physical system. On the channel (from supervisory controller and actuator), the
one hand, the need for the interconnection of all things command issued by a supervisor may not be effectively or
51
successfully accepted by the plant. Hence, in consideration controllability w.p. 1 are given for NFSM on the basis
of the complex evolutionary behavior, the modeling and of the transition probability matrix.
analyzing of communication delays and packet losses has
The remaining sections are structured as follows: Section
given rise to new challenges. Meanwhile, the utilization
2 brings in some basic notations and knowledge about
of communication networks in the control loops have
the NFSM and matrix STP. Section 3 focuses on the key
also considerable theoretical and practical significance for
results of this paper, including the algebraic representation
NFSM.
for NFSM with random channel packet losses, and the
From the analysis of the current literature, many
validation criteria for the existence of controllability. In
researches on NFSM almost always concentrate on the
section 4, one typical simulation example is illustrated to
channel delays, such as centralized control [8] and
validate the proposed results. Finally, in Section 5, we
decentralized control [9], robust control [10] and distributed
summarize the whole paper and give a brief prospect of the
failure prognosis [11] and so on. In recent years, Lin
future study.
[12] investigated the bounded communication losses in the
control and observation channel from the perspective of
2. Basic notations and preliminaries
supervisory control, and obtained the existence conditions
for networked supervisor, i.e., network observability and
• N + is said to be the set of all positive integers.
controllability need to be met simultaneously. Up to
• Mm×l is expressed as the set of m × l-dimensional real
now, There are fewer works to address the issues on the
matrices.
controllability problem with communication losses in the
• A(i, j) ∈ Mm×l represents the element with the i-th row
control channel, except our previous work [13], which has
and j-th column of matrix M.
solved the reachability problem with communication losses
• The j-th column of A ∈ Mm×l can be expressed as
from a switched perspective. However, in [13], we assumed
Col j (M), and furthermore, the set of all columns is
that the communication losses occur determinately, i.e.,
Col(M).
the probability of random channel packet losses from the W
• is termed as the logical ‘OR’ operation.
controller to the actuator is not considered, which lacks the
• 1n = [1, 1, · · · , 1].
strict quantitative analysis for random packet losses. | {z }
n
Inspired by the methods in [13], i.e., matrix STP method, • δkn is the k-th column of identity matrix In , k ∈
this paper continues to study the influence of random packet {1, 2, · · · , n}.
losses on controllability in the control channel of NFSM. • ∆n := {δ1n , δ2n , · · · , δnn }.
We refer to the work in [8], and assume that all the events • ν = [ν1 , ν2 , . . . , νn ]T ∈ Rn is said to be sub-stochastic
are disabled by default, and then, construct another finite logical vector if every element is satisfied that νi ≥ 0
and ni=1 νi 6 1; Especially, it is defined as stochastic
P
state machine affected by packet losses, where the set of
logical vector if ni=1 νi = 1.
P
transitions with communication losses are disabled and the
predecessor of such transitions will remain unchanged with • The set of n-dimensional sub-stochastic logical vector
respect to the lost events. To characterize the dynamics is denoted by L̃ns , making Lns as the set of n-
with random channel packet losses, the state estimator dimensional stochastic logical vector.
is introduced by utilizing a stochastic variable, which is • E[y] is called the expected value of sub-stochastic or
assumed to obey the Bernoulli random binary distribution. stochastic logical vector y.
Some key contributions include the following points: • Σ∗ denotes the set, which is composed of all finite-
length and non-empty strings on the finite set Σ.
• The matrix expression of NFSM with random channel
packet losses in control is proposed by an algebraic In the classical FSM, the distance from controller to
state space method. actuator is assumed to be close, and the supervisory
• The validation criteria for the reachability and command can be transmitted to the plant instantaneously
Mathematical Modelling and Control Volume 3, Issue 1, 50–6052
through the control channel. In general, the finite state δm+2
mn , · · · , δmn
(n−1)m+2
, · · · , δm
mn , · · · , δmn ],
nm
machine can be briefly written as G = Q, Σ, f, q0 , and
j
it consists of four-tuple, including the set of events Σ and where δmn is termed as the j-th column of identity
states Q from the initial state q0 , the (partial) state transition matrix Imn
function f : Q × Σ → Q, generating all possible transitions
(2) Given a matrix of arbitrary dimension X ∈ Mm×n , if it
f = {(q, σ, q0 ) : f (q, σ) = q0 }. Additionally, the transition
is supposed that X (0) := In , then for k ∈ N + , the STP
function f can be generalized over Σ∗ by an iterative way.
power is denoted by
Due to the relatively far transmission distance, it is noted
that packet losses between the supervisory controller and X (k) := | }.
n ··· n X
X n X {z (2.2)
actuator will be introduced inevitably. According to the k
actual situation of engineering system modeling and the
information transmission network, those transitions affected The STP-based algebraic state space method can be
by the actuators that are far away from the supervisor may described as the following two procedures:
be lost at random. We define such transitions as the set
• Firstly, by using the matrix STP, the logical variables
fl ⊆ f , and the remaining transitions are supposed to not
involved in the system are represented by a vector of
be absolutely lost, which is written as ful = f − fl . That is,
finite dimensions (xi ∼ δin );
the set of all transitions can be divided into two parts, i.e.,
• Secondly, the logical dynamic system is transferred into
f = fl ∪ ful . In view of the above condition, a NFSM can be
a bilinear system on the finite state set. The bilinear
characterized as A = {A, fl }.
representation is used to study the process of the logical
At the beginning of the 21-st century, a novel algebraic
dynamic system.
framework is proposed on the basis of the matrix semi-
tensor product, which transforms the logic dynamic system In accordance with the modeling process of algebraic
into algebraic representation and extends the application state space method, a NFSM A can be redefined as
scope of the classical linear state space approach. Since Q = {q1 , q2 , · · · , qn } and Σ = {σ1 , σ2 , · · · , σm }, and these
then, the STP-based algebraic state space method has variables are further set as qi = δin (i ∈ {1, 2 . . . , n})
inspired and generated an abound of excellent works in and σk = δkm (k ∈ {1, 2 . . . , m}). Meanwhile, for
related areas, such as logic control networks [14–20], multi- the convenience, we list some variable symbols used
agent systems [21], networked evolution games [22–25] and later. qul (ς) = [qul 1 (ς), q2 (ς), · · · , qn (ς)]
ul ul T
(ql (ς) =
discrete event systems [26–28] and so on [29]. First of all, [ql1 (ς), ql2 (ς), · · · , qln (ς)]T ) is expressed as the vector form of
for the convenience of understanding, we give some related state reached at t steps with (no) control packet loss, and
i (ς) = 1 (qi (ς) = 1) is defined if, and only, if there is
qul
definitions and properties of matrix STP. l
a transition that qi is reachable from q0 with (no) control
Definition 2.1 ( [30]). Given two matrices of arbitrary
packet loss; u(ς) = [u1 (ς), u2 (ς), · · · , um (ς)]T is termed as
dimensions H ∈ Mm×n , D ∈ M p×q , the corresponding STP
the event vector on Σ, and uk (ς) = 1 if σk is enabled at ς
(‘n’) of H and D is defined as:
steps. In the next section, we present the model dynamics
H n D := (H ⊗ Ir/n )(D ⊗ Ir/p ), (2.1) and controllability analysis of NFSM with packet losses
based on the representations.
where r is called the least common multiple of n and p.
3. Controllability under random channel packet losses
(1) For χ ∈ Mm×1 , ϕ ∈ Mn×1 , the pseudo-commutativity of
χ and ϕ is satisfied that ϕ n χ = Ψ[m,n] n χ n ϕ, where
3.1. Models for random channel packet losses
Ψ[m,n] is a constructed swap matrix, defined as:
In networked control systems, channel packet loss is
Ψ[m,n] = [δ1mn , δm+1
mn , δmn , · · ·
2m+1
, δ(n−1)m+1
mn , δ2mn , inevitable due to the restricted bandwidth, especially
Mathematical Modelling and Control Volume 3, Issue 1, 50–6053
in control (from supervisory controller to some remote hypothesized to be transmitted through the control channel,
actuators), which will lead to the occurrence of failure and Figure 1 shows the state transition relationship among
control. Firstly, given a NFSM A = (A, fl ), various states. Otherwise, the newly constructed finite
state machine is represented as Figure 2. Note that the
Assumption 3.1. Before we present the main results, some
added self-loops in Figure 2 are represented by the dashed
assumptions are made in the following:
arrows on account of the communication losses, which can
1. Each event in Σ is disabled by default, and is only be obtained by substituting (q1 , σ2 , q3 ), (q3 , σ1 , q2 ) with
allowed to occur if it is enabled by a control command. (q1 , σ2 , q1 ), (q3 , σ1 , q3 ), respectively.
2. The communication losses are random, i.e., the 1
q1 q2 1
transitions in fl may or may not be communicated in
control.
2 1 1, 2
On the one hand, if the transitions in set fl are assumed
to be communicated successfully in the control channel, q3 q4
the algebraic representation can be constructed in a manner 2
similar to that described in the reference [13], and the Figure 1. NFSM with no control packet loss.
dynamics of NFSM is shown as a discrete-time bi-linear
form in the following form:
1
2 q1 q2 1
q (ς + 1) = F n u(ς) n q (ς),
ul ul ul
(3.1)
where F ul is called the state transition structure matrix with 1, 2
no control packet loss, and it is specifically defined as
2
1 q3 q4
F ul := [F1ul , F2ul , · · · , Fmul ] ∈ Mn×mn , (3.2)
Figure 2. NFSM with the control channel packet
where Fkul ∈ Mn×n is defined as: loss.
j
1, δn ∈ f (δin , δkm )
ul
=
Fk( j,i) (3.3) According to (3.2) and (3.3), we define a new structure
0, otherwise.
matrix F l := [F1l , F2l , · · · , Fml ] ∈ Mn×mn for the constructed
On the other hand, if the transitions in fl fails to be NFSM, and the dynamics with control packet losses will be
communicated to the plant, such transitions are disabled and iterated according to the following equation:
will be permitted to occur until the transitions are not lost
in a later step. In particular, for (q, σ, q0 ) ∈ fl , not any ql (ς + 1) = F l n u(ς) n ql (ς). (3.4)
transition labeled by σ is enabled from state q to q . In this 0
situation, state q will remain unchanged in itself with respect However, the dynamics expressed by (3.1) and (3.4)
to event σ under communication losses, which will alter the can characterize both cases with complete communication
transition structure of NFSM. In order to characterize such losses, and the case with no packet loss. In practice, the
changed transitions, we construct a new finite state machine, losses in the control channel are random, i.e., the transitions
in which the transitions will be replaced with a self-loop in fl may or may not be communicated in the control
if (q, σ, q0 ) ∈ fl is not transmitted to the performer. Let’s channel. So as to cope with the random packet losses, we
illustrate the above process with a simple example. introduce a state estimation with random channel packet
losses, which can be depicted as follows:
Example 3.1. Consider a NFSM A = (A, fl ), where fl =
{(q1 , σ2 , q3 ), (q3 , σ1 , q2 )}, and if the transitions in fl are q(ς) = (1 − γ)qul (ς) + γql (ς), (3.5)
Mathematical Modelling and Control Volume 3, Issue 1, 50–6054
where γ is a random variable on behalf of the packet loss In fact, the aforementioned theorem tells us that the
rate, which is independent of system state. Note that q(ς) NFSM with random channel packet losses can be captured
is also stochastic because it merely refers to the source of by a special probabilistic finite state machine [31]. Next,
randomness from γ, which is indeed a random variable, by means of the matrix expression in Theorem 3.1,
then the expectation of q(ς) will be implemented. In this controllability of NFSM with random channel packet losses
work, it is assumed that the communication losses may will be investigated.
happen according to the following Bernoulli random binary
Remark 3.1. In the previous work [13], we discussed the
distribution:
influence of arbitrary packet loss on the reachability, and
a novel theoretical framework is proposed to analyze the
Prob{γ = 1} = E[γ] = λ (3.6)
robustness of reachable state from a switched perspective.
Prob{γ = 0} = 1 − E[γ] = 1 − λ, (3.7)
Here, the controllability is studied by the matrix STP, but
we concentrates on its analysis from a stochastic view. We
where λ ∈ [0, 1) is a constant and represents the probability
firmly believe that the models proposed in this paper and
that any transition in fl will be lost.
[13] provide a very important basis for studying the control
In the following theorem, the dynamics of NFSM with
problem with network packet loss.
random channel packet losses can be established.
Theorem 3.1. Given a NFSM A = (A, fl ) with channel 3.2. Controllability analysis
packet loss rate λ, the evolutionary dynamics can be In this subsection, we continue to investigate the
described by the following stochastic logical equation: controllability of NFSM (network controllability), where
random packet losses are considered. Here, the definition of
E[q(ς + 1)] = F u(ς)E[q(ς)], (3.8)
controllability with random channel packet losses are given
as follows:
where F = (1 − λ)F ul + λF l is termed as the probabilistic
transition structure matrix (PTSM), and E[q(ς)] ∈ Lns Definition 3.1. Given a NFSM A = (A, fl ) with channel
denotes the expected value of the reachable state q(ς) and packet loss rate λ,
E[q(0)] = q0 .
(1) from the initial state q0 = δin , the target state q j = δnj
Proof. : For any state q in the transition (q, σ, q0 ) ∈ ful , is reachable at the k-th step w.p. 1 (k-th reachable) if
q(ς) = q (ς) = q (ς) is satisfied. According to (3.5), (3.6)
ul l
there is an event string s = σl0 σl1 . . . , σlk−1 ∈ Σ∗ so that
and (3.7), we can obtain that Prob{q(k) = q j | q(0) = q0 } = 1.
(2) the set of all k-th reachable states from q0 is denoted
E[q(ς + 1)] =E[(1 − γ)qul (ς + 1) + γql (ς + 1)]
by Rk (q0 ). Furthermore, R(q0 ) is said to be the set
=E[(1 − γ)]E[qul (ς + 1)] + E[γ]E[ql (ς + 1)]
of all states reachable w.p. 1 from q0 , and R(q0 ) =
=(1 − E(γ))F ul n u(ς) n E[qul (ς)]+
∪i∈N + Ri q0 .
E[γ]F l n u(ς) n E[ql (ς)]
(3) system is said to be controllable w.p. 1 from q0 if
=((1 − λ)F + λF )u(ς)E[q(ς)].
ul l
(3.9)
R(q0 ) = ∆n .
For state q in the transition (q, σ, q0 ) ∈ fl , q(ς) = qul (ς) = Let us first introduce the notion of k-th transition
l
q (ς) may be not satisfied. Generally, it is not hard to get that probability matrix (TPM) with some sequence of events.
E[q(ς + 1)] equals to E[(1 − γ)qul (ς + 1) + γ0] or equals to Consider a NFSM with random channel packet losses.
E[(1 − γ)0 + γq (ς + 1)]. To sum up, E[q(ς + 1)] = ((1 −
l
Suppose a specified input u(ς) = δkm , Theorem 3.1 means
λ)F ul + λF l )u(ς)E[q(ς)] represents the state transitions of E[q(ς + 1)] = F δkm E[q(ς)] = Fk E[q(ς)]; in this case,
NFSM with random packet loss rate. Fk ∈ Mm×n is called the first step TPM from the current state
Mathematical Modelling and Control Volume 3, Issue 1, 50–6055
q(ς) to the next q(ς + 1) with the input u(ς), and rewritten as Theorem 3.3. Given a NFSM A = (A, fl ) with channel
P1 . Denote k-th TPM by Pk ∈ Mn×n , whose (i, j) entry is packet loss rate λ,
Prob{q(t + k) = q j | q(ς) = qi }, i.e., the probability from the
(1) the target state q j = δnj is k-th reachable from q0 = δin
state vector q(ς) = δin to q(ς + k) = δnj under a sequence of
iff
c=0 u(ς +c) that is executed. By using the matrix STP
events nk−1
Lk( j,i) = 1; (3.13)
and Theorem 3.1, Pk can be equivalently calculated through
the iterative computation, and established in the following (2) the target state q j = δnj is reachable w.p. 1 from q0 = δin
proposition. iff there is a positive integer α, such that
Theorem 3.2. Given a NFSM A = (A, fl ) with a sequence of α
_
events s = σl0 σl1 . . . , σlk−1 ∈ Σ∗ , then the k-th TPM satisfies. Lk( j,i) = 1; (3.14)
k=1
Pk = D(k) nk−1
c=0 δm ,
lc
(3.10)
(3) the NFSM is controllable w.p. 1 from q0 = δin iff the
where D(k) = (F Ψ[n,m] )(k) Ψ[mk ,n] ∈ Mn×nmk , and Ψ[n,m] can
positive integer α satisfies the following condition
be referred to the pseudo-commutativity.
α
_
Proof. Based on the matrix expression (3.8), we have the Coli (Lk ) = 1n . (3.15)
following result: k=1
E[q(1)] = F Ψ[n,m] E[q(0)]u(0) Proof. (1) ‘if’ part: According to the definition of operator
D E
E[q(2)] = F Ψ[n,m] E[q(1)]u(1) h•i in Eq. (3.12), Lk( j,i) = 1, i.e., D(k) = 1 iff there is a
( j,i)
= (F Ψ[n,m] ) E[q(0)]u(0)u(1)
(2) positive integer β ∈ {1, 2, . . . , mk }, such that (D(k)
β )( j,i) = 1.
.. According to (3.11), it means that a sequence of events
. s = σl0 σl1 . . . , σlk−1 ∈ Σ∗ exists that (D(k) nc=0 δm )( j,i) = 1,
k−1 lc
E[q(k)] = F Ψ[n,m] E[q(k − 1)]u(k − 1) i.e., Pk( j,i) = 1 is satisfied. Then, an input string s =
= (F Ψ[n,m] ) E[q(0)]
(k)
nk−1
c=0 δlmc σl0 σl1 . . . , σlk−1 ∈ Σ can satisfy the condition Prob{q(k) =
∗
(3.11) q j | q(0) = q } = 1.
0
= (F Ψ[n,m] )(k) Ψ[mk ,n] nk−1
c=0 δm E[q(0)].
lc
‘only if’ part: If state q j = δnj is k-th reachable from q0 =
If (F Ψ[n,m] )(k) Ψ[mk ,n] is abbreviated as D(k) , then, k-th TPM δi , we prove that Lk = 1 is satisfied. Based on the matrix
n ( j,i)
Pk with a sequence of events s = σl0 σl1 . . . , σlk−1 expression (3.8) and Theorem 3.2, the condition shows that
∈ Σ∗ can be calculated as D(k) nk−1
c=0 δm .
lc
there is a sequence of events s = σl0 σl1 . . . , σlk−1 ∈ Σ∗ , such
Theorem 3.2 reveals that the matrix D(k) ∈ Mn×nmk s that δnj = E[x(k)] = Pk δin , i.e., Pk( j,i) = 1, which implies
consists of the current state transition probability and the that β ∈ {1, 2, . . . , mk } exists, such that (D(k)
β )( j,i) = 1. Then
D E
event strings with respect to the reachable paths of A, which clearly, L( j,i) = ( D )( j,i) = 1 is tenable.
k (k)
provides an important basis for system control analysis (2) ‘if’ part: we suppose that there is a positive integer α
and design. Here, let us go further to split D(k) into mk such that
blocks, and D(k)
i ∈ Mn×n , i.e., D(k) = [D(k)
1 , D2 , . . . , Dmk ].
(k) (k) α
_
To characterize the controllability of NFSM with random Lk( j,i) = L1(j,i) ∨ L2(j,i) ∨ · · · ∨ Lα( j,i) = 1,
k=1
channel packet losses, an operator h•i is introduced and
defined by and the existence of k ∈ {1, 2, . . . , α} makes the formula
Lk( j,i) = 1 true. Based on Eq. (3.13), the target state q j = δnj
mk
is k-th reachable w.p. 1 from q0 = δin . i.e., state q j = δnj is
D E _
1 ∨ D2 ∨ · · · ∨ Dmk =
Lk := D(k) := D(k) (k) (k)
i . (3.12)
D(k)
i=1 reachable w.p. 1 from q0 = δin .
Finally, the main results to validate the criterion are ‘only if’ part: If state q j = δnj is reachable w.p. 1 from
derived based on the above preliminaries. q0 = δin , based on the result mentioned above, we can obtain
Mathematical Modelling and Control Volume 3, Issue 1, 50–6056
Wα
that Lk( 1j,i) = 1, Lk( 2j,i) = 1, · · · , or Lk( τj,i) = 1. Thus, k=1 Lk( j,i) = Example 4.1. Considering a networked flexible
1 can be arrived at, where α = max{k1 , k2 , . . . , kτ }. manufacturing system, it is characterized as a FSM
(3) ‘if’ part: If there is a positive integer α such that A = Q, Σ, f, q0 , and its corresponding evolution
α
_ graph is depicted in Figure 3, where Q = {q1 , · · · , q9 }
Coli (Lk ) = Coli (L1 ) ∨ Coli (L2 ) ∨ · · · ∨ Coli (Lα ) = 1n , and Σ = {σ1 , · · · , σ4 } denote the different states and
k=1
signal indicators. Due to the relatively far transmission
it is satisfied that Lk(1,i)
1
= 1, Lk(2,i)
2
= 1, · · · , L(n,i)
kτ
= 1,distance, it is assumed that the occurrences in fl =
where ki ∈ {1, 2, . . . , α}. Based on the aforementioned result, {(q1 , σ2 , q7 ), (q7 , σ3 , q1 ), (q2 , σ3 , q8 ), (q8 , σ4 , q2 ), (q3 , σ1 , q9 ),
state q j = δnj is k j -th reachable w.p. 1 from q0 = δin , (q9 , σ3 , q3 )} is communicated with channel packet loss rate
i.e., the k j -step reachable set w.p. 1 Rk j (q0 ) = q j , and λ = 0.25. The following graph (see Figure 4) represents the
R(q0 ) = ∪ j≤α Rk j q0 = ∆n . Henceforth, the NFSM system state transition diagram with packet losses.
with random packet loss is thought to be controllable w.p. 1
3
from q0 .
1 2
‘only if’ part: If the system is controllable w.p. 1 from q2 q5 q3
q =
0
δin , it is obvious that the reachable states w.p. 1 from 2
2
q satisfy R(q0 ) = ∆n . According to Eq. (3.13), we can
0 4 3
1
3 1
derive that Lk(1,i)
1
= 1, Lk(2,i)
2
= 1, · · · , Lk(n,i)
τ
= 1. In brief,
Wα 3 q8 q4 q9
k=1 Coli (L ) = 1n , where α = max{k1 , k2 , . . . , kτ }.
k
2
3 4
Remark 3.2. In this paper, the impact of random 2 2
channel packet losses on the controllability of NFSM is
2
systematically studied. We discover that the dynamics of q6 q1 q7
3 3
NFSM with random channel packet losses can be expressed
1
as a probabilistic finite state machine by utilizing the
state estimation, and such dynamics take into account the Figure 3. The evolution graph of A, or the state
probability λ of random channel packet losses from the transition diagram without any packet losses in fl .
controller to actuator, which provides us an important model
for quantitative analysis. 3
1 2
Remark 3.3. On a broader note, the underlying evolution 3 q2 q5 q3 1
of the states seems analogous to that of a discrete-time 2
2
finite Markov chain, where the transition probabilities 1
4
depend on the packet loss. The results in this paper
3 q8 q4 2
q9 3
are equivalent to the irreducibility of the Markov chain.
However, we put more emphasis on the controllable paths 3 4
2 2
from the initial state. With the assistance of matrix STP,
2
the algebraic representation of NFSM with channel packet
q6 q1 q7
loss rate λ is established, and the verification criteria for 3
3
the controllability w.p. 1 are derived by the k-th transition 1
probability matrix Pk . Figure 4. State transition diagram with packet
losses, which is similar with Figure 2. Also note
4. Numerical example that dashed arrows represent the lost transitions.
In this section, we utilize a typical example to validate the When taking the packet loss rate into account, the
theoretical results in Section 3. evolution dynamics on NFSM can be termed as E[q(ς +
Mathematical Modelling and Control Volume 3, Issue 1, 50–6057
3，
1
0 0 0 0 0 0 0.25 1 0
2，
1，
1 1 0 0.75 0 0 0 1 0 0 0
3 , 0.75 q2 q3 1 , 0.75
q5
0 0 0 0 0 0 0 0 0.25
2，
1 1，
1
2，
1 0 0 0 0 0 0 0 0 0
1，
0.25
4 , 0.25 , 0.25
3 0.25
F3 =
3
0 0 0 0 1 0 0 0 0
3，
1 q8 q4 q9 3 , 0.75 1 0 0 0 0 0 0 0 0
4 , 0.75 2，
1
0 0 0 0 0 0 0.75 0 0
3，
1 4，
1
2，
1 2 , 0.75 2，
1 0 0.25 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0.75
2 , 0.25
q6 q1 q7
3，
1 3 , 0.25
3 , 0.75
1，
1
0 0 0 0 0 0 0 0 1
Figure 5. The resulting constructed state diagram
0 0 0 0 0 0 0 0.25 0
with channel packet loss rate λ = 0.25.
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1)] = F u(ς)E[q(ς)], and its corresponding PTSM can be F4 =
0 0 0 0 0 0 0 0 0
written as F = {F1 , F2 , F3 , F4 }, which is calculated based
0 0 0 0 0 0 0 0 0
on Theorem 3.1. The resulting constructed state diagram is
0 0 0 0 0 0 0 0 0
shown as Figure 5 by adding the channel packet loss rate λ = 0 0 0 0 0 0 0 0.75 0
0.25. In this manufacturing system, some transactions are 0 0 0 0 0 0 0 0 0
always assigned to accomplish important tasks, for example,
from interface states q2 , q5 to q1 . Next, we will show how to According to Theorem 3.3, when k = 3, the reachability
verify the controllability w.p. 1. matrix with random packet loss rate kµ=1 L(µ) can be
W
obtained in the following:
1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 1 0.3 1 1 1 1 1 1
0 0 0.75 1 0 0 0 0 0
1 0.6 0 0 0 0.8 0.3 1 1
0 0 0 0 0 0 0 0 0
0.1 1 1 0.8 1 1 0.3 0.3 0.2
F1 =
0 1 0 0 0 0 0 0 0
3
1 1 1 1 1 1 0 0.3 0.3
0 0 0 0 0 0 0 0 0
_
L(µ) =
1 1 1 1 1 1 0.3 0.3 0.3
0 0 0 0 0 0 0 0 0 µ=1
1 0.3 0.3 1 0 1 1 1 1
0 0 0 0 0 0 0 0 0
0.3 0.1 0.3 0 0.3 0.7 0.3 0.3
0 0 0.25 0 0 0 0 0 0
1 0.2 0 0 0 0.3 0.3 1 1
0.3 1 1 0.8 1 1 0.8 0.3 0.6
0.75 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
L(µ)
(1,5) = L(µ)
(1,2) = 1, state q1 = δ9 is
W3 W3 1
0 0 0 0 0 0 0 0 0 Notice that µ=1 µ=1
reachable w.p. 1 from q2 = δ29 and q5 = δ59 , respectively.
0 1 0 0 1 0 0 0 0
F2 = 0 0 1 0 0 0 0 0 0 When considering system information security issues,
some states are supposed to be safe, such as q1 = δ19 and the
0 0 0 0 0 0 0 0 0
0.25 0 0 0 0 0 0 0 0 state will be labeled as a target state, which is required to be
0 0 0 0 0 1 0 0 0
arrived from all states. Through some iterative calculations,
if k = 4, the reachability matrix with random packet loss rate
0 0 0 1 0 0 1 0 0
Mathematical Modelling and Control Volume 3, Issue 1, 50–6058
W4
µ=1 L(µ) is computed as: Conflict of interest
1 1 1 1 1 1 1 1 1 The authors declare that there are no potential conflicts of
1 0.6 0.3 1 0 1 1 1 1 interests.
1 1 1 1 1 1 0.3 0.3 0.3
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